Hello,
In the first method,it must calculate x[k-1] again when every time it
calculates x[k].So the recursion is slow.But in the second one,x[k-1]
is saved in the memory,namely a.So it can be called straightforward
from memory,need not calculate again.
as regards the limit of this recursion for k->Infinite,you can try
this:
FixedPoint[0.25(1 + # + #^2 + #^3) &, 0]